The International CAPM Redux
Francesca Brusa (Oxford)
Tarun Ramadorai (Oxford and CEPR)
Adrien Verdelhan (MIT Sloan and NBER)
March 2015
Aggregate Foreign Equity Holdings
(as % of global GDP)
0.35
Assets As a Percentage of World GDP
0.3
0.25
0.2
0.15
0.1
0.05
0
1970
1975
1980
1985
1990
Portfolios of Foreign Equity Assets
1995
2000
2005
2010
De Jure Index of Financial Openness
Motivation
I
I
Large increase in financial integration over the past 30 years.
I
Aggregate foreign equity holdings, as a % of global GDP: from
3% in the 1980s to approximately 30% in 2011.
I
Foreign equity holdings of the U.S.: $6 trillion.
What are the expected returns on those foreign equity
holdings?
I
Finance logic:
I
I
How to measure bad times?
I
I
If an asset’s return tends to be low in bad times, the asset is
risky. Its expected return should be above the risk-free rate.
Capital Asset Pricing Model (CAPM): Aggregate market
return
How to compare returns?
I
Foreign equity returns are in foreign currencies
Currency Risk
I
Exchange rate risk should matter. . .
I
I
Yet, empirical evidence is difficult to obtain.
I
I
When investors invest abroad, but consume at home, and
domestic and foreign purchasing powers differ.
International CAPM of Dumas and Solnik (1995): world stock
market return + three bilateral exchange rates (Yen, Mark,
and Pound).
Recent findings: Two risk factors account for a large share of
systematic changes in bilateral exchange rates
I
Idea: Instead of using raw bilateral exchange rates, use
currency risk factors that summarize their systematic variation.
This Paper
1. Three factors describe the time-series of foreign equity
benchmark returns
I
Smaller difference between expected and realized returns than
in the competitors’ models (World CAPM, International
CAPM, and Fama-French factors)
2. Those three factors account for a large share of foreign equity
mutual funds’ and macro/emerging hedge funds’ returns
3. A simple reduced-form model to study the interaction between
currency and equity risk
Realized vs Predicted Excess Returns
Actual mean excess return (%)
World CAPM
International CAPM
3
3
2
2
1
1
0
0
−1
−1
0
1
2
−1
−1
3
Actual mean excess return (%)
Fama−French Four Factors
3
2
2
1
1
0
0
Aggr. Market
0
1
2
Predicted excess return (%)
Value
Growth
1
2
3
CAPM Redux
3
−1
−1
0
−1
−1
3
Small
Big
0
1
2
Predicted excess return (%)
Carry Portfolios
3
Dollar Portfolios
Literature
I World CAPM:
I
I
Sharpe (1964) and Lintner (1965) to Stehle (1977), Solnik (1974), . . .
Korajczyk and Viallet (1989), Harvey (1991), Chan, Karolyi and Stulz
(1992), Bekaert and Harvey (1995), Karolyi and Stulz (1996)
I International CAPM:
I
I
Solnik (1974), Sercu (1980), Stulz (1981), Adler and Dumas (1983), . . .
Bekaert and Hodrick (1992), Ferson and Harvey (1993), Ferson and
Harvey (1994), Bekaert (1995, 1996), Dumas and Solnik (1995), De
Santis and Gerard (1998), Harvey, Solnik, and Zhou (2002)
I New equity factors and frictions:
I
Chaieb and Errunza (2007), Bekaert, Harvey and Lundblad (2007); Hou,
Karolyi, and Kho (2011), Karolyi, Lee and van Dijk (2012), Karolyi and
Wu (2014), Malkhozov, Mueller, Vedolin, and Venter (2014)
I Currency drivers:
I
order flows (e.g., Evans and Lyons, 2002, and Froot and Ramadorai,
2005) and risk factors (e.g., Lustig, Roussanov and Verdelhan, 2011,
2014, Menkhoff, Sarno, Scheming, and Schrimpf, 2012, Maggiori, 2012,
Lettau, Maggiori, and Weber, 2013, and Verdelhan, 2014)
Road map
I
I
I
Empirical evidence:
I
Equity benchmark indices
I
Hedge funds and mutual funds
Theoretical framework:
I
Intuition
I
Replication in a reduced-form model
Conclusion
Empirical Asset Pricing
Asset Pricing
I
When the law of one price holds and investors can form
portfolios freely, there exists a SDF Mt+1 that prices any
i
return Rt+1
:
i
Et Mt+1 Rt+1
=1
I
The same condition holds for the risk-free rate Rf .
I
Assuming that the returns and SDF are lognormal, the Euler
equation implies:
i )
covt (mt+1 , rt+1
1
i
i
) =−
vart (mt+1 )
Et rt+1
− rf ,t + vart (rt+1
2
vart (mt+1 ) | {z }
|
{z
}
Λt
βti
where lower letters denote logs.
I
What is the SDF Mt+1 ?
I
If M is an excess return, its market price of risk Λ should be
equal to its mean.
Equity Literature Summary
I
World CAPM:
WMKTt+1 = world stock market return in U.S. dollars
e,i,$
ri,t+1
− rf ,t
I
i
αi + βWMKT
[WMKTt+1 − rf ,t ] + t+1
International CAPM: + 3 bilateral exchange rates
e,i,$
ri,t+1
− rf ,t
I
=
=
i
αi + βWMKT
[WMKTt+1 − rf ,t ]
+
i
GBP
i
JPY
i
EUR
βGBP
rt+1
+ βJPY
rt+1
+ βEUR
rt+1
+ t+1
Three/four Fama-French factor models: size, value, and
momentum anomalies
e,i,$
ri,t+1
− rf ,t
=
i
αi + βWMKT
[WMKTt+1 − rf ,t+1 ]
+
i
i
i
βSMB
SMBt+1 + βHML
HMLt+1 + βWML
WMLt+1 + t+1
Currency Literature Summary
I
Carry trade risk: sort currencies by their short-term interest
rates
E (R e )
1
2
-3.17
-0.46
Portfolios
3
4
1.26
1.80
5
6
2.29
4.48
Source: Lustig, Roussanov, and Verdelhan, 2011.
I
“Dollar” risk: sort currencies by their exchange rate betas,
long if average interest rate > U.S. interest rate, short
otherwise
E (R e )
1
2
1.31
2.86
Portfolios
3
4
3.87
5.13
Source: Verdelhan, 2014.
5
6
4.99
7.14
Intuition
I
Three risk factors: Equity, Dollar, and Carry.
1
i
i
Et rt+1 − rf ,t + vart (rt+1 ) = βti,1 Λ1t + βti,2 Λ2t + βti,3 Λ3t .
2
I
One could ignore currency risk and focus on equity risk only if
all the quantities and prices of risk were in sync.
1
f
i
i
Et rt+1
− rf ,t + vart (rt+1
) = βti, Λet .
2
I
No reason to expect so.
This Paper: International CAPM Redux
I
Three-factor model:
e,i,$
Carry
i
i
Dollar
i
ri,t+1
− rf ,t = αi + βLWMKT
[LWMKTt+1 − rf ,t ] + βDollar
rt+1
+ βCarry
rt+1
+ t+1
I
Factors:
1. Global equity (LWMKT): built from local currency equity
returns.
2. Dollar: average excess return of a U.S. investor going long
foreign Treasury Bills.
3. Carry: average excess return of a U.S. investor going long high
interest rate currencies and short low interest rate currencies.
I
Estimation: time-varying betas obtained on 60-month rolling
windows.
Data and Estimation
I
Test assets: from 46 developed and emerging countries,
spanning value, growth, and country index returns from
1/1976 to 12/2013.
I
Risk factors: global equity factor (built from local currency
equity returns) and two currency factors: Dollar and Carry
I
Estimation: time-varying betas obtained on rolling windows
Test Assets (I/II)
Developed Markets
20
15
%
10
5
0
−5
−10
Portugal
Ireland
Japan
Austria
Israel
New Zealand
Italy
Greece
United States
Spain
Canada
Germany
United Kingdom
France
Singapore
Belgium
Countries
Switzerland
Denmark
Netherlands
Norway
Australia
Finland
Sweden
China Hong Kong
Aggr. Market
Growth
Small
Big
Value
Growth
Aggr. Market
Value
Big
Asset type
Small
Test Assets (II/II)
Emerging Markets
30
25
20
%
15
10
5
0
−5
China
Morocco
Taiwan
Malaysia
Philippines
South Korea
India
South Africa
Czech Republic
Thailand
Hungary
Chile
Egypt
Colombia
Peru
Indonesia
Countries
Poland
Mexico
Turkey
Russia
Brazil
China
Morocco
Taiwan
Aggr. Market
Growth
Small
Big
Value
Growth
Aggr. Market
Value
Big
Asset type
Small
Expected vs Realized Equity Excess Returns
I Since factors are excess returns, the market prices of risk should be the means of
the risk factors (no-arbitrage condition)
ET
I
I
e,i,$
rt+1
− rf ,t
e,i,$
rt+1 − rf ,t
=
αi + β i Ft+1 + t+1
=
αi + β i ET (Ft+1 )
For each asset and each rolling window:
I
Estimate the beta on rolling windows, from t − 59 to t:
βt
I
Estimate the price of risk as the mean of the risk factor
over the P
same period, from t − 59 to t:
1
λt = 60 59
i=0 Ft−i
I
Form expected equity excess returnsas the product
of
e,i,$
the quantities and prices of risk: Et rt+1 − rf ,t = βt λt
Average across time and compare expected to realized excess
returns
Time-varying Factor Betas: Developed Markets
Exposure to LWMKT
Exposure to Dollar
Exposure to Carry
Australia
Austria
Belgium
Canada
Denmark
Finland
France
Germany
Greece
Hong Kong
Ireland
Israel
Italy
Japan
Netherlands
New Zealand
Norway
Portugal
Singapore
Spain
Sweden
Switzerland
United Kingdom
United States
−1
0
1
βLWMKT
2
3
−1
0
Average
1
βDollar
2
3
Min/Max
−1
0
1
βCarry
2
3
Time-varying Factor Betas: Emerging Markets
Exposure to LWMKT
Exposure to Dollar
Exposure to Carry
Brazil
Chile
China
Colombia
Czech Republic
Egypt
Hungary
India
Indonesia
Malaysia
Mexico
Morocco
Peru
Philippines
Poland
Russia
South Africa
South Korea
Taiwan
Thailand
Turkey
−4
−2
0
2
βLWMKT
4
6
−4
−2
0
Average
βDollar
2
4
6
Min/Max
−4
−2
0
βCarry
2
4
6
Realized vs Predicted Excess Returns
Actual mean excess return (%)
World CAPM
International CAPM
3
3
2
2
1
1
0
0
−1
−1
0
1
2
−1
−1
3
Actual mean excess return (%)
Fama−French Four Factors
3
2
2
1
1
0
0
Aggr. Market
0
1
2
Predicted excess return (%)
Value
Growth
1
2
3
CAPM Redux
3
−1
−1
0
−1
−1
3
Small
Big
0
1
2
Predicted excess return (%)
Carry Portfolios
3
Dollar Portfolios
Robustness Checks
I Other comparisons of the pricing errors:
I
Histograms and kernel density estimates of rolling alphas
I
Rolling mean absolute alphas
I Even if the market prices of risk are estimated (Fama-McBeth procedure), the
CAPM Redux still compares favorably:
I
the market prices of risk are positive and significant for our three factors
I
equity risk is priced for the CAPM, International CAPM, and FF models,
but not the other factors (except GBP); the prices of risk appear further
removed from the mean of the risk factors.
I Subsamples:
I
time-windows (increasing sizes starting from 1976 or backwards from
2013)
I
test assets (country aggregates, developed vs emerging markets)
I
length of the rolling windows
Kernel Density of Rolling Alphas
0.6
Kernel density
0.4
0.2
0
−5
−3
−1
World CAPM
1
3
Rolling alphas
Fama−French Four Factors
5
7
CAPM Redux
9
Rolling Mean Absolute Alphas
1.6
1.4
Mean Absolute Alphas
1.2
1
0.8
0.6
0.4
0.2
Jan−81
Aug−85
WCAPM
Mar−90
Oct−94
Jun−99
Jan−04
Fama−French Four Factors
Aug−08
Apr−13
CAPM Redux
Correlation Between Equity Returns & Currency Factors
LOC Equity Returns and Carry factor
1
0.8
0.8
0.6
0.6
0.4
0.2
Corr(Ret, Carry)t
Corr(Ret, Dollar)t
LOC Equity Returns and Dollar factor
1
0.4
0.2
0
0
−0.2
−0.2
−0.4
−0.4
Jan−81 Aug−85 Apr−90 Nov−94 Jun−99 Jan−04 Sep−08 Apr−13
Jan−81 Aug−85 Apr−90 Nov−94 Jun−99 Jan−04 Sep−08 Apr−13
Mutual and Hedge Fund Returns
Hedge Funds and Mutual Funds
I
Fund’s exposure to global systematic factors:
i
Rt+1
= αi + β i Carryt+1 + γ i Dollart+1 + δ i LWMKTt+1 + εit+1 ,
i
where Rt+1
denotes the monthly realized return of fund i.
I
Data:
I
MFs: U.S. “Foreign Equity Funds” from CRSP.
I
HFs: International “Macro” and “Emerging” Funds from
Ramadorai (2013) and Patton, Ramadorai and Streatfield
(2013).
I
Estimation: time-varying betas obtained on rolling windows.
I
Sample period: November 1990 (MF)/January 1994 (HF) to
April 2013.
Mutual Funds’ Significant Exposure to Currency Factors
Number
of funds
1200
600
0
1996
1998
2000
2002
2004
2006
2008
2010
2012
1996
1998
2000
2002
2004
2006
2008
2010
2012
1996
1998
2000
2002
2004
2006
2008
2010
2012
1
Dollar
Factor
0.8
0.6
0.4
0.2
0
1
Carry
Factor
0.8
0.6
0.4
0.2
0
“Macro/Emerging” Hedge Funds’ Significant Exposure to Currency Factors
Number
of funds
400
200
0
2000
2002
2004
2006
2008
2010
2012
2000
2002
2004
2006
2008
2010
2012
2000
2002
2004
2006
2008
2010
2012
1
Dollar
Factor
0.8
0.6
0.4
0.2
0
1
Carry
Factor
0.8
0.6
0.4
0.2
0
Statistical Significance of Positive HF and MF’s Alphas
Hedge Funds
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
ECDF
ECDF
Mutual Funds
1
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
0
1
2
3
4
5
0
1
t(α>0)
World CAPM
2
3
t(α>0)
Int. CAPM
Fama−French Four Factors
CAPM Redux
4
5
Theoretical Framework
Intuition
I Recall that:
Et
i
rt+1
− rf ,t +
i )
covt (mt+1 , rt+1
1
i
vart (rt+1
) =−
vart (mt+1 )
2
vart (mt+1 )
|
{z
}
|
{z
}
Λt
βti
I Expected currency excess returns when markets are complete and shocks are
gaussian (Bekaert, 1996, Bansal, 1997, Backus, Foresi, and Telmer, 2001):
i
Et rxt+1
=
i
rfi ,t − rf ,t − Et ∆st+1
=
1
1
i
Vart (mt+1 ) − Vart (mt+1
)
2
2
I Failure of the uncovered interest rate parity (U.I.P.) implies that expected
currency excess returns are time-varying.
I SDF must be heteroscedastic, and thus market prices of risk must be
time-varying.
Framework
I
Assume that financial markets are complete (thus log changes
in exchange rates are differences in log SDF)
I
Start from the law of motion of each SDF, which depends on
country-specific and global shocks
I
Assume that each country-aggregate dividend growth rate
depends on the same shocks as the SDF
I
Two key assumptions:
I
Prices of risk are time-varying (time-varying currency risk
premia)
I
One global shock affects all SDF in the same way (role for a
pure equity risk factor)
Reduced-Form Model with Endogenous Exchange Rates
I
In the tradition of Frachot (1996), Backus, Foresi and Telmer
(2001), Brennan and Xia (2006)
I
Exponentially affine SDFs in complete financial markets
i
−mt+1
=
+
i
zt+1
=
w
zt+1
=
i
∆dt+1
=
+
q
i
α + χzti + γzti ut+1
q
q
p
g
w
c
τ ztw + δ i ztw ut+1
+ κzti ut+1
,
+ ωztw ut+1
q
i
(1 − φ)θ + φzti − σ zti ut+1
,
p
w
w w
w w
w
(1 − φ )θ + φ zt − σ
ztw ut+1
q
i
µD + ψzti + ψw ztw + σD zti ut+1
q
p
g
g
c,i p w c
w
w
σD
ztw ut+1
+ σD
zti ut+1
+ σD
zt ut+1
i
w , ug , uc
where ut+1
, ut+1
t+1 t+1 are i.i.d, mean-zero, variance-one Gaussian shocks.
I
Similar model studied in Lustig, Roussanov, and Verdelhan (2011,
2014) and in Verdelhan (2014) but without the global “equity”
c
shocks ut+1
Model Solution
I
Simple and tractable reduced-form model:
I
I
I
Interest rates, exchange rates, and price-dividend ratios in
closed-forms, as well as realized and expected equity and
currency excess returns, and equity and currency risk factors.
Implications:
I
The world stock market return in U.S. dollars depends on all
global shocks.
I
No role for bilateral exchange rates in theory
I
But in the model the market price of global equity risk depends
on several state variables.
I
Those state variables are unknown to the econometrician.
I
Quantities and market prices of risk evolve at different
frequencies.
Role for currency factors in practice, even in simulated data
Realized vs Predicted Excess Returns: Simulated Data
World CAPM
CAPM Redux
7
Realized average excess returns (%)
Realized average excess returns (%)
7
6.5
6
5.5
5
4.5
4
3.5
3
6.5
6
5.5
5
4.5
4
3.5
3
3 3.5 4 4.5 5 5.5 6 6.5 7
3 3.5 4 4.5 5 5.5 6 6.5 7
Predicted excess returns (%)
Predicted excess returns (%)
Conclusion
I
Three factors (Global Equity, Dollar, and Carry) describe the
time-series of foreign equity benchmark returns.
I
Predicted and realized equity excess returns are closer than in
the competitors’ models.
I
Currency risk matters for foreign equity mutual fund and
“macro/emerging” hedge fund returns.
I
Interaction between currency and equity risk in a simple
reduced-form model:
I
Time-variation in quantities and prices of risk is key.